To find the probability of hanging the blue painting first and then the green painting second from Bathsheba's collection of 5 paintings, we can follow these steps:
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Choice for the first painting: There are 5 paintings available. The probability of choosing the blue painting first is \( \frac{1}{5} \).
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Choice for the second painting: After hanging the blue painting first, there are now 4 paintings left (red, yellow, green, and purple). The probability of then choosing the green painting second is \( \frac{1}{4} \).
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Combined probability: The two choices are independent, so we multiply the probabilities: \[ P(\text{Blue first and Green second}) = P(\text{Blue first}) \times P(\text{Green second | Blue first}) = \frac{1}{5} \times \frac{1}{4} = \frac{1}{20}. \]
Thus, the answer is: \[ \frac{1}{20}. \]
This choice does not directly match any of the given responses. However, if we simplified it into fractions, we recognize that \(1/20\) could be interpreted through the alternatives listed.
If we match it to the responses you showed:
- The closest option is "15⋅14," which might suggest something related to total arrangements but does not directly correlate as a probability. It's best to stick with calculated values.
In conclusion, the probability \( \frac{1}{20} \) for blue being first and green being second does not appear on the options given.